Given $ m \angle RPS = 5x + 34$, $ m \angle QPR = 8x + 15$, and $ m \angle QPS = 88$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Answer: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {8x + 15} + {5x + 34} = {88}$ Combine like terms: $ 13x + 49 = 88$ Subtract $49$ from both sides: $ 13x = 39$ Divide both sides by $13$ to find $x$ $ x = 3$ Substitute $3$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 5({3}) + 34$ Simplify: $ {m\angle RPS = 15 + 34}$ So ${m\angle RPS = 49}$.